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Theorem isumclim3 13897
Description: The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that  j must not occur in  A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
isumclim3.1  |-  Z  =  ( ZZ>= `  M )
isumclim3.2  |-  ( ph  ->  M  e.  ZZ )
isumclim3.3  |-  ( ph  ->  F  e.  dom  ~~>  )
isumclim3.4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumclim3.5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
Assertion
Ref Expression
isumclim3  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Distinct variable groups:    A, j    j, k, M    ph, j, k   
j, Z, k    j, F
Allowed substitution hints:    A( k)    F( k)

Proof of Theorem isumclim3
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumclim3.3 . . 3  |-  ( ph  ->  F  e.  dom  ~~>  )
2 climdm 13695 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 201 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 sumfc 13852 . . . 4  |-  sum_ m  e.  Z  ( (
k  e.  Z  |->  A ) `  m )  =  sum_ k  e.  Z  A
5 isumclim3.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
6 isumclim3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
7 eqidd 2472 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
8 isumclim3.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
9 eqid 2471 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
108, 9fmptd 6061 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
1110ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
125, 6, 7, 11isum 13862 . . . 4  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
134, 12syl5eqr 2519 . . 3  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ) ) )
14 seqex 12253 . . . . . . 7  |-  seq M
(  +  ,  ( k  e.  Z  |->  A ) )  e.  _V
1514a1i 11 . . . . . 6  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  A ) )  e.  _V )
16 isumclim3.5 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
17 fzssuz 11865 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1817, 5sseqtr4i 3451 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
19 resmpt 5160 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2018, 19ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2120fveq1i 5880 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
22 fvres 5893 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2321, 22syl5reqr 2520 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2423sumeq2i 13842 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
25 sumfc 13852 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
2624, 25eqtri 2493 . . . . . . . 8  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
27 eqidd 2472 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
28 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
2928, 5syl6eleq 2559 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpl 464 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
31 elfzuz 11822 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3231, 5syl6eleqr 2560 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  m  e.  Z )
3330, 32, 11syl2an 485 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
3427, 29, 33fsumser 13873 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3526, 34syl5eqr 2519 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ k  e.  ( M ... j
) A  =  (  seq M (  +  ,  ( k  e.  Z  |->  A ) ) `
 j ) )
3616, 35eqtr2d 2506 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
375, 15, 1, 6, 36climeq 13708 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3837iotabidv 5574 . . . 4  |-  ( ph  ->  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
39 df-fv 5597 . . . 4  |-  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ) )  =  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )
40 df-fv 5597 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4138, 39, 403eqtr4g 2530 . . 3  |-  ( ph  ->  (  ~~>  `  seq M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4213, 41eqtrd 2505 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
433, 42breqtrrd 4422 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    |` cres 4841   iotacio 5551   ` cfv 5589  (class class class)co 6308   CCcc 9555    + caddc 9560   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810    seqcseq 12251    ~~> cli 13625   sum_csu 13829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830
This theorem is referenced by:  esumcvg  28981
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